再论一类二次系统的无界双中心周期环域的POincare分支

本文再一次讨论了具有双曲线与赤道弧为边界的双中心周期环域的二次系统的Poincare分支，并构造出了此系统出现极限环的(0，3)分布或出现一个三重极限环的具体例子．     

Some Global Optimization Problems on Stiefel Manifolds

Optimization on Stiefel manifolds was discussed by Rapcsák in earlier papers, and some global optimization methods were considered and tested on Stiefel manifolds. In the paper, test functions are given with known global optimum points and their optimal function values. A restriction, which leads to a discretization of the problem is suggested, which results in a problem equivalent to the well-known assignment problem.     

A Rigorous Lower Bound for the Optimal Value of Convex Optimization Problems

In this paper, we consider the computation of a rigorous lower error bound for the optimal value of convex optimization problems. A discussion of large-scale problems, degenerate problems, and quadratic programming problems is included. It is allowed that parameters, whichdefine the convex constraints and the convex objective functions, may be uncertain and may vary between given lower and upper bounds. The error bound is verified for the family of convex optimization problems which correspond to these uncertainties. It can be used to perform a rigorous sensitivity analysis in convex programming, provided the width of the uncertainties is not too large. Branch and bound algorithms can be made reliable by using such rigorous lower bounds.     

Characteristic Elements of Unimodular -lattices

Characteristic elements have been useful in the classification of unimodular lattices over the integers. This article gives an explicit formula for characteristic elements of a lattice in terms of a basis for the lattice and the dual of that basis.     

Approximation of the Quadratic Set Covering problem

We study in this article the polynomial approximation properties of the Quadratic Set Covering problem. This problem, which arises in many applications, is a natural generalization of the usual Set Covering problem. We show that this problem is very hard to approximate in the general case, and even in classical subcases (when the size of each set or when the frequency of each element is bounded by a constant). Then we focus on the convex case and give both positive and negative approximation results. Finally, we tackle the unweighted version of this problem.     

Energy and Quadratic Invariants Preserving Methods for Hamiltonian Systems with Holonomic Constraints

We introduce a new class of parametrized structure--preserving partitioned Runge-Kutta ($\alpha$-PRK) methods for Hamiltonian systems with holonomic constraints. The methods are symplectic for any fixed scalar parameter $\alpha$, and are reduced to the usual symplectic PRK methods like Shake-Rattle method or PRK schemes based on Lobatto IIIA-IIIB pairs when $\alpha=0$. We provide a new variational formulation for symplectic PRK schemes and use it to prove that the $\alpha$-PRK methods can preserve the quadratic invariants for Hamiltonian systems subject to holonomic constraints. Meanwhile, for any given consistent initial values $(p_{0}, q_0)$ and small step size $h>0$, it is proved that there exists $\alpha^*=\alpha(h, p_0, q_0)$ such that the Hamiltonian energy can also be exactly preserved at each step. Based on this, we propose some energy and quadratic invariants preserving $\alpha$-PRK methods. These $\alpha$-PRK methods are shown to have the same convergence rate as the usual PRK methods and perform very well in various numerical experiments.     

Quadratic cone cutting surfaces for quadratic programs with on–off constraints

We study the convex hull of a set arising as a relaxation of difficult convex mixed integer quadratic programs (MIQP). We characterize the extreme points of the convex hull of the set and the extreme points of its continuous relaxation. We derive four quadratic cutting surfaces that improve the strength of the continuous relaxation. Each of the cutting surfaces is second-order-cone representable. Via a shooting experiment, we provide empirical evidence as to the importance of each inequality type in improving the relaxation. Computational results that employ the new cutting surfaces to strengthen the relaxation for MIQPs arising from portfolio optimization applications are promising.     

复杂结构优化选型的计算方法

本文为复杂结构优化选型提供了一个比较实用的计算方法:根据原有结构型式和设计经验来建立优化选型计算模型;对目标函数和约束条件作数学上近似处理后,利用Kuhn-Tucker条件和Lagrange乘子,把选型问题转化为求解典型二次规划问题;引进加权因子和利用改进后的变尺度法使计算过程简化、可靠,而且精度较高.     

Explicit Solutions of Quadratic FBSDEs Arising From Quadratic Term Structure Models

We provide explicit solutions of certain forward-backward stochastic differential equations (FBSDEs) with quadratic growth. These particular FBSDEs are associated with quadratic term structure models of interest rates and characterize the zero-coupon bond price. The results of this paper are naturally related to similar results on affine term structure models of Hyndman (Math. Financ. Econ. 2(2):107–128, 2009) due to the relationship between quadratic functionals of Gaussian processes and linear functionals of affine processes. Similar to the affine case a sufficient condition for the explicit solutions to hold is the solvability in a fixed interval of Riccati-type ordinary differential equations. However, in contrast to the affine case, these Riccati equations are easily associated with those occurring in linear-quadratic control problems. We also consider quadratic models for a risky asset price and characterize the futures price and forward price of the asset in terms of similar FBSDEs. An example is considered, using an approach based on stochastic flows that is related to the FBSDE approach, to further emphasize the parallels between the affine and quadratic models. An appendix discusses solvability and explicit solutions of the Riccati equations.